For example, if I take log(1), (in any base at all), I know immediately that that's gotta be 0. But the key to remember is that the input of logs has to be the output of some exponential, so there's a few things you get automatically. It might be a bit trickier with log to begin with, because you aren't used to it. How do you know how to graph those functions? You plug in values and plot them out, then connect them smoothly. Log(x) is a function just as nice as sin(x) or x^3. To graph log functions, you aren't really solving for anything. Do you mean how they find the values? Like, if I took log_2(8), how does it "find" that the answer is 3? Well, how do you know that sqrt(16) = 4? It's really the same idea. I'm not entirely sure what you mean by "the way logs work". Its similar to how the cube root is the inverse of the cube.
Its the result of asking the question: if I have a value, say a = b^x, and I know a and b, how do I find x? Clearly, there's some relationship between a and x, and log_b(a) is exactly that number. Well, logs are inverses to exponents by definition.
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